Critical phenomena in semi-infinite systems. II. Mean-field theory

Abstract

The critical behavior of a continuous spin system in a semi-infinite sample is studied for all values of the extrapolation length $\lambda$ using mean-field theory. A new transition, which we have called the extraordinary transition, is found for $\lambda <0\mathrm{}$ in which the bulk orders at a temperature below the surface ordering temperature. In this paper we have calculated the magnetic susceptibilities $\chi \mathrm{}\left(z\right)\mathrm{}$ and $\chi (z, z)$ and the correlation function $\Gamma (\overrightarrow{\mathrm{x}}, {\overrightarrow{\mathrm{x}}}^{\prime })$ at all the phase transitions. We have used these results to compute the various $\gamma$ and $\eta$ exponents and to study the scaling relations introduced by Binder and Hohenberg.